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Iterative Techniques in Matrix Algebra
Jacobi & Gauss-Seidel Iterative Techniques II
Numerical Analysis (9th Edition)
R L Burden & J D Faires
Beamer Presentation Slidesprepared byJohn Carroll
Dublin City University
c 2011 Brooks/Cole, Cengage Learning
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
G S id l M h d G S id l Al i h C R l I i
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 3 / 38
G S id l M th d G S id l Al ith C R lt I t t ti
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi MethodA possible improvement to the Jacobi Algorithm can be seen by
re-considering
x (k )i = 1a ii
n j =1 j =i
−a ij x (k −1) j
+ b i
, for i = 1, 2, . . . , n
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 4 / 38
Gauss Seidel Method Gauss Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi MethodA possible improvement to the Jacobi Algorithm can be seen by
re-considering
x (k )i = 1a ii
n j =1 j =i
−a ij x (k −1) j
+ b i
, for i = 1, 2, . . . , n
The components of x(k −1) are used to compute all the
components x (k )i of x(k ).
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 4 / 38
Gauss Seidel Method Gauss Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi MethodA possible improvement to the Jacobi Algorithm can be seen by
re-considering
x (k )i = 1a ii
n j =1 j =i
−a ij x (k −1) j
+ b i
, for i = 1, 2, . . . , n
The components of x(k −1) are used to compute all the
components x (k )i of x(k ).
But, for i > 1, the components x (k )1 , . . . , x
(k )i −1 of x
(k ) have already
been computed and are expected to be better approximations to
the actual solutions x 1, . . . , x i −1 than are x (k −1)1 , . . . , x
(k −1)i −1 .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 4 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Instead of using
x (k )i =
1
a ii
n j =1 j =i
−a ij x
(k −1) j
+ b i
, for i = 1, 2, . . . , n
it seems reasonable, then, to compute x (k )i using these most recently
calculated values.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 5 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss Seidel Method Gauss Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Instead of using
x (k )i =
1
a ii
n j =1 j =i
−a ij x
(k −1) j
+ b i
, for i = 1, 2, . . . , n
it seems reasonable, then, to compute x (k )i using these most recently
calculated values.
The Gauss-Seidel Iterative Technique
x (k )i =
1
a ii
−
i −1 j =1
(a ij x (k )
j ) −n
j =i +1
(a ij x (k −1)
j ) + b i
for each i = 1, 2, . . . , n .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 5 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss Seidel Method Gauss Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutionsto
10x 1 − x 2 + 2x 3 = 6
−x 1 + 11x 2 − x 3 + 3x 4 = 25
2x 1 − x 2 + 10x 3 − x 4 = −113x 2 − x 3 + 8x 4 = 15
,
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 6 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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g g p
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutionsto
10x 1 − x 2 + 2x 3 = 6
−x 1 + 11x 2 − x 3 + 3x 4 = 25
2x 1 − x 2 + 10x 3 − x 4 = −113x 2 − x 3 + 8x 4 = 15
,
starting with x = (0, 0, 0, 0)t
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 6 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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g g p
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutionsto
10x 1 − x 2 + 2x 3 = 6
−x 1 + 11x 2 − x 3 + 3x 4 = 25
2x 1 − x 2 + 10x 3 − x 4 = −113x 2 − x 3 + 8x 4 = 15
,
starting with x = (0, 0, 0, 0)t and iterating until
x(k )
−x(k −1)
∞x(k )∞
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The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutionsto
10x 1 − x 2 + 2x 3 = 6
−x 1 + 11x 2 − x 3 + 3x 4 = 25
2x 1 − x 2 + 10x 3 − x 4 = −113x 2 − x 3 + 8x 4 = 15
,
starting with x = (0, 0, 0, 0)t and iterating until
x(k )
−x(k −1)
∞x(k )∞
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The Gauss-Seidel Method
Solution (1/3)
For the Gauss-Seidel method we write the system, for each
k = 1, 2, . . . as
x (k )1 = 110x (k −1)2 − 15
x (k −1)3 + 35
x (k )2 =
1
11x (k )1 +
1
11x (k −1)3 −
3
11x (k −1)4 +
25
11
x (k )3 = −
1
5 x (k )1 +
1
10 x (k )2 +
1
10 x (k −1)4 −
11
10
x (k )4 = −
3
8x (k )2 +
1
8x (k )3 +
15
8
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 7 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method
Solution (2/3)
When x(0) = (0, 0, 0, 0)t , we havex(1) = (0.6000, 2.3272, −0.9873, 0.8789)t .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 8 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method
Solution (2/3)
When x(0) = (0, 0, 0, 0)t , we havex(1) = (0.6000, 2.3272, −0.9873, 0.8789)t . Subsequent iterations givethe values in the following table:
k 0 1 2 3 4 5
x (k )1 0.0000 0.6000 1.030 1.0065 1.0009 1.0001
x (k )
2
0.0000 2.3272 2.037 2.0036 2.0003 2.0000
x (k )
3 0.0000 −0.9873 −1.014 −1.0025 −1.0003 −1.0000
x (k )4 0.0000 0.8789 0.984 0.9983 0.9999 1.0000
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 8 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method
Solution (3/3)
Becausex(5) − x(4)∞
x(5)∞ =
0.0008
2.000 = 4 × 10−4
x(5) is accepted as a reasonable approximation to the solution.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 9 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method
Solution (3/3)
Becausex(5) − x(4)∞
x(5)∞ =
0.0008
2.000 = 4 × 10−4
x(5) is accepted as a reasonable approximation to the solution.
Note that, in an earlier example, Jacobi’s method required twice as
many iterations for the same accuracy.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 9 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form,
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 10 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form, multiply both sides of
x (k )i = 1a ii
−i −1 j =1
(a ij x (k ) j ) −
n j =i +1
(a ij x (k −1) j ) + b i
by a ii and collect all k th iterate terms,
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 10 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form, multiply both sides of
x (k )i = 1
a ii
−
i −1 j =1
(a ij x (k ) j ) −
n j =i +1
(a ij x (k −1) j ) + b i
by a ii and collect all k th iterate terms, to give
a i 1x (k )1 + a i 2x (
k )2 + · · · + a ii x (
k )i = −a i ,i +1x (
k −
1)i +1 − · · · − a in x (
k −
1)n + b i
for each i = 1, 2, . . . , n .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 10 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations (Cont’d)
Writing all n equations gives
a11x(k)1 = −a12x
(k−1)2 − a13x
(k−1)3 − · · · − a1nx
(k−1)n + b1
a21x(k)
1 + a
22x(k)
2 = −a
23x(k−1)
3 − · · · − a
2nx(k−1)
n + b
2
.
.
.
an1x(k)1 + an2x
(k)2 + · · · + annx
(k)n = bn
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 11 / 38
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The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations (Cont’d)
Writing all n equations gives
a11x(k)1 = −a12x
(k−1)2 − a13x
(k−1)3 − · · · − a1nx
(k−1)n + b1
a21x(k)
1 + a
22x(k)
2 = −a
23x(k−1)
3 − · · · − a
2nx(k−1)
n + b
2
.
.
.
an1x(k)1 + an2x
(k)2 + · · · + annx
(k)n = bn
With the definitions of D , L, and U given previously, we have the
Gauss-Seidel method represented by
(D − L)x(k ) = U x(k −1) + b
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The Gauss-Seidel Method: Matrix Form
(D − L)x(k ) = U x(k −1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k ) finally gives
x(k ) = (D − L)−1U x(k −1) + (D − L)−1b, for each k = 1, 2, . . .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 12 / 38
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The Gauss-Seidel Method: Matrix Form
(D − L)x(k ) = U x(k −1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k ) finally gives
x(k ) = (D − L)−1U x(k −1) + (D − L)−1b, for each k = 1, 2, . . .
Letting T g = (D − L)−1U and cg = (D − L)
−1b, gives the Gauss-Seidel
technique the form
x(k ) = T g x(k −1) + cg
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 12 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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The Gauss-Seidel Method: Matrix Form
(D − L)x(k ) = U x(k −1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k ) finally gives
x(k ) = (D − L)−1U x(k −1) + (D − L)−1b, for each k = 1, 2, . . .
Letting T g = (D − L)−1U and cg = (D − L)
−1b, gives the Gauss-Seidel
technique the form
x(k ) = T g x(k −1) + cg
For the lower-triangular matrix D − L to be nonsingular, it is necessaryand sufficient that a ii = 0, for each i = 1, 2, . . . , n .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 12 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 13 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 14 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
INPUT the number of equations and unknowns n ;
the entries a ij
, 1 ≤
i , j ≤
n of the matrix A;
the entries b i , 1 ≤ i ≤ n of b;
the entries XO i , 1 ≤ i ≤ n of XO = x(0);
tolerance TOL;
maximum number of iterations N .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 14 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
INPUT the number of equations and unknowns n ;
the entries a ij
, 1 ≤
i , j ≤
n of the matrix A;
the entries b i , 1 ≤ i ≤ n of b;
the entries XO i , 1 ≤ i ≤ n of XO = x(0);
tolerance TOL;
maximum number of iterations N .
OUTPUT the approximate solution x 1, . . . , x n or a message
that the number of iterations was exceeded.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 14 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k ≤ N ) do Steps 3–6:
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k ≤ N ) do Steps 3–6:
Step 3 For i = 1, . . . , n
set x i = 1
a ii
−i −1
j =1
a ij x j −n
j =i +1
a ij XO j + b i
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 15 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
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Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k ≤ N ) do Steps 3–6:
Step 3 For i = 1, . . . , n
set x i = 1
a ii
−i −1
j =1
a ij x j −n
j =i +1
a ij XO j + b i
Step 4 If ||x − XO||
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Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k ≤ N ) do Steps 3–6:
Step 3 For i = 1, . . . , n
set x i = 1
a ii
−i −1
j =1
a ij x j −n
j =i +1
a ij XO j + b i
Step 4 If ||x − XO||
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Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k ≤ N ) do Steps 3–6:
Step 3 For i = 1, . . . , n
set x i = 1
a ii
−i −1
j =1
a ij x j −n
j =i +1
a ij XO j + b i
Step 4 If ||x − XO||
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Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k ≤ N ) do Steps 3–6:
Step 3 For i = 1, . . . , n
set x i = 1
a ii
−i −1
j =1
a ij x j −n
j =i +1
a ij XO j + b i
Step 4 If ||x − XO||
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Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that a ii = 0, for eachi = 1, 2, . . . , n .
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
G S id l It ti Al ith
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Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that a ii = 0, for eachi = 1, 2, . . . , n . If one of the a ii entries is 0 and the system isnonsingular, a reordering of the equations can be performed so
that no a ii = 0.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Ga ss Seidel Iterati e Algorithm
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Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that a ii = 0, for eachi = 1, 2, . . . , n . If one of the a ii entries is 0 and the system isnonsingular, a reordering of the equations can be performed so
that no a ii = 0.
To speed convergence, the equations should be arranged so that
a ii is as large as possible.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss Seidel Iterative Algorithm
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Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that a ii = 0, for eachi = 1, 2, . . . , n . If one of the a ii entries is 0 and the system isnonsingular, a reordering of the equations can be performed so
that no a ii = 0.
To speed convergence, the equations should be arranged so that
a ii is as large as possible.
Another possible stopping criterion in Step 4 is to iterate until
x(k ) − x(k −1)
x(k )
is smaller than some prescribed tolerance.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss Seidel Iterative Algorithm
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Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that a ii = 0, for eachi = 1, 2, . . . , n . If one of the a ii entries is 0 and the system isnonsingular, a reordering of the equations can be performed so
that no a ii = 0.
To speed convergence, the equations should be arranged so that
a ii is as large as possible.
Another possible stopping criterion in Step 4 is to iterate until
x(k ) − x(k −1)
x(k )
is smaller than some prescribed tolerance.
For this purpose, any convenient norm can be used, the usual
being the l ∞ norm.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 16 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
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Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 17 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
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Convergence Results for General Iteration Methods
Introduction
To study the convergence of general iteration techniques, we need
to analyze the formula
x(k ) = T x(k −1) + c, for each k = 1, 2, . . .
where x(0) is arbitrary.
The following lemma and the earlier Theorem on convergent
matrices provide the key for this study.
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 18 / 38
Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
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Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies ρ(T )
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Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies ρ(T )
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Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies ρ(T )
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Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies ρ(T )
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Convergence Results for General Iteration Methods
Proof (2/2)Let
S m = I + T + T 2 + · · · + T m
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Convergence Results for General Iteration Methods
Proof (2/2)Let
S m = I + T + T 2 + · · · + T m
Then
(I − T )S m = (1 + T + T 2 + · · · + T m ) − (T + T 2 + · · · + T m +1) = I − T m +1
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Convergence Results for General Iteration Methods
Proof (2/2)Let
S m = I + T + T 2 + · · · + T m
Then
(I − T )S m = (1 + T + T 2 + · · · + T m ) − (T + T 2 + · · · + T m +1) = I − T m +1
and, since T is convergent, the Theorem on convergent matrices
implies that
limm →∞(I − T )S m = limm →∞(I − T m +1) = I
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g
Proof (2/2)Let
S m = I + T + T 2 + · · · + T m
Then
(I − T )S m = (1 + T + T 2 + · · · + T m ) − (T + T 2 + · · · + T m +1) = I − T m +1
and, since T is convergent, the Theorem on convergent matrices
implies that
limm →∞(I − T )S m = limm →∞(I − T m +1) = I
Thus, (I − T )−1 = limm →∞ S m = I + T + T 2 + · · · =
∞ j =0 T
j
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g
Theorem
For any x(0) ∈ IRn , the sequence {x(k )}∞k =0 defined by
x(k )
= T x(k −1)
+c
, for each k ≥ 1
converges to the unique solution of
x = T x + c
if and only if ρ(T )
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g
Proof (1/5)First assume that ρ(T )
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g
Proof (1/5)First assume that ρ(T )
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Proof (1/5)First assume that ρ(T )
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Proof (1/5)
First assume that ρ(T )
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Proof (1/5)
First assume that ρ(T )
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Proof (1/5)
First assume that ρ(T )
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Proof (2/5)The previous lemma implies that
limk →∞
x(k ) = lim
k →∞
T k x(0) +
∞
j =0
T j
c
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Proof (2/5)The previous lemma implies that
limk →∞
x(k ) = lim
k →∞
T k x(0) +
∞
j =0
T j
c
= 0 + (I − T )−1c
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Proof (2/5)The previous lemma implies that
limk →∞
x(k ) = lim
k →∞
T k x(0) +
∞
j =0
T j
c
= 0 + (I − T )−1c
= (I − T )−1c
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Proof (2/5)The previous lemma implies that
limk →∞
x(k ) = lim
k →∞
T k x(0) +
∞
j =0
T j
c
= 0 + (I − T )−1c
= (I − T )−1c
Hence, the sequence {x(k )} converges to the vector x ≡ (I − T )−1cand x = T x + c.
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Proof (3/5)
To prove the converse, we will show that for any z ∈ IRn , we havelimk →∞ T
k z = 0.
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Proof (3/5)
To prove the converse, we will show that for any z ∈ IRn , we havelimk →∞ T
k z = 0.
Again, by the theorem on convergent matrices, this is equivalentto ρ(T )
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Proof (3/5)
To prove the converse, we will show that for any z ∈ IRn , we havelimk →∞ T
k z = 0.
Again, by the theorem on convergent matrices, this is equivalentto ρ(T )
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Proof (3/5)
To prove the converse, we will show that for any z ∈ IRn , we havelimk →∞ T
k z = 0.
Again, by the theorem on convergent matrices, this is equivalentto ρ(T )
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Proof (3/5)
To prove the converse, we will show that for any z ∈ IRn , we havelimk →∞ T
k z = 0.
Again, by the theorem on convergent matrices, this is equivalentto ρ(T )
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Proof (4/5)
Also,
x − x(k ) = (T x + c) −
T x(k −1) + c
= T x − x(k −1)
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Proof (4/5)
Also,
x − x(k ) = (T x + c) −
T x(k −1) + c
= T x − x(k −1)
sox − x(k ) = T
x − x(k −1)
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Proof (4/5)
Also,
x − x(k ) = (T x + c) −
T x(k −1) + c
= T x − x(k −1)
sox − x(k ) = T
x − x(k −1)
= T 2x − x(k −2)
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Proof (4/5)
Also,
x − x(k ) = (T x + c) −
T x(k −1) + c
= T x − x(k −1)
sox − x(k ) = T
x − x(k −1)
= T 2x − x(k −2)
=
...
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Proof (4/5)
Also,
x − x(k ) = (T x + c) −
T x(k −1) + c
= T x − x(k −1)
sox − x(k ) = T
x − x(k −1)
= T 2x − x(k −2)
=
...
= T k x − x(0)
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Proof (4/5)
Also,
x − x(k ) = (T x + c) −
T x(k −1) + c
= T x − x(k −1)
sox − x(k ) = T
x − x(k −1)
= T 2x − x(k −2)
=
...
= T k x − x(0)
= T k z
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Proof (5/5)
Hence
limk →∞
T k z = limk →∞
T k x − x(0)
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Proof (5/5)
Hence
limk →∞
T k z = limk →∞
T k x − x(0)
= limk →∞
x − x(k )
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Proof (5/5)
Hence
limk →∞
T k z = limk →∞
T k x − x(0)
= limk →∞
x − x(k )
= 0
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Proof (5/5)
Hence
limk →∞
T k z = limk →∞
T k x − x(0)
= limk →∞
x − x(k )
= 0
But z ∈ IRn was arbitrary, so by the theorem on convergentmatrices, T is convergent and ρ(T ) < 1.
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Corollary
T
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Corollary
T
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Corollary
T
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Corollary
T
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1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
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Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k ) = T j x
(k −1) + c j and
x(k ) = T g x(k −1) + cg
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Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k ) = T j x
(k −1) + c j and
x(k ) = T g x(k −1) + cg
using the matrices
T j = D −1(L + U ) and T g = (D − L)
−1U
respectively.
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Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k ) = T j x
(k −1) + c j and
x(k ) = T g x(k −1) + cg
using the matrices
T j = D −1(L + U ) and T g = (D − L)
−1U
respectively. If ρ(T j ) or ρ(T g ) is less than 1, then the correspondingsequence {x(k )}∞k =0 will converge to the solution x of Ax = b.
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Example
For example, the Jacobi method has
x(k ) = D −1(L + U )x(k −1) + D −1b,
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Example
For example, the Jacobi method has
x(k ) = D −1(L + U )x(k −1) + D −1b,
and, if {x(k )}∞k =0
converges to x,
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Example
For example, the Jacobi method has
x(k ) = D −1(L + U )x(k −1) + D −1b,
and, if {x(k )}∞k =0
converges to x, then
x = D −1(L + U )x + D −1b
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Example
For example, the Jacobi method has
x(k ) = D −1(L + U )x(k −1) + D −1b,
and, if {x(k )}∞k =0
converges to x, then
x = D −1(L + U )x + D −1b
This implies that
D x = (L + U )x + b and (D − L − U )x = b
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Example
For example, the Jacobi method has
x(k ) = D −1(L + U )x(k −1) + D −1b,
and, if {x(k )}∞k =0
converges to x, then
x = D −1(L + U )x + D −1b
This implies that
D x = (L + U )x + b and (D − L − U )x = b
Since D − L − U = A, the solution x satisfies Ax = b.
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The following are easily verified sufficiency conditions for convergence
of the Jacobi and Gauss-Seidel methods.
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The following are easily verified sufficiency conditions for convergence
of the Jacobi and Gauss-Seidel methods.
Theorem
If A is strictly diagonally dominant, then for any choice of x(0), both the
Jacobi and Gauss-Seidel methods give sequences {x(k )}∞k =0 thatconverge to the unique solution of Ax = b.
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Is Gauss-Seidel better than Jacobi?
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Is Gauss-Seidel better than Jacobi?
No general results exist to tell which of the two techniques, Jacobi
or Gauss-Seidel, will be most successful for an arbitrary linear
system.
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S
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(Stein-Rosenberg) Theorem
If a ij ≤ 0, for each i = j and a ii > 0, for each i = 1, 2, . . . , n , then oneand only one of the following statements holds:
(i) 0 ≤ ρ(T g ) < ρ(T j )
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(Stein-Rosenberg) Theorem
If a ij ≤ 0, for each i = j and a ii > 0, for each i = 1, 2, . . . , n , then oneand only one of the following statements holds:
(i) 0 ≤ ρ(T g ) < ρ(T j )
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Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 ≤ ρ(T g ) < ρ(T j )
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Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 ≤ ρ(T g ) < ρ(T j )
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Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 ≤ ρ(T g ) < ρ(T j )
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Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 ≤ ρ(T g ) < ρ(T j )
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Eigenvalues & Eigenvectors: Convergent Matrices
Theorem
The following statements are equivalent.
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g q
(i) A is a convergent matrix.
(ii) limn →∞ An = 0, for some natural norm.
(iii) limn →∞ An = 0, for all natural norms.
(iv) ρ(A)
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Let g ∈ C [a , b ] be such that g (x ) ∈ [a , b ], for all x in [a , b ]. Suppose, inaddition, that g exists on (a , b ) and that a constant 0
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Corrollary to the Fixed-Point Convergence Result
If g satisfies the hypothesis of the Fixed-Point Theorem then
|p n − p | ≤ k n
1 − k |p 1 − p 0|
Return to the Corollary to the Convergence Theorem for General Iterative Methods
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